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BCH codes have been studied for over fifty years and widely employed inconsumer devices, communication systems, and data storage systems. However, thedimension of BCH codes is settled only for a very small number of cases. Inthis paper, we study the dimensions of BCH codes over finite fields with threetypes of lengths $n$, namely $n=q^m-1$, $n=(q^m-1)/(q-1)$ and $n=q^m+1$. Fornarrow-sense primitive BCH codes with designed distance $\delta$, weinvestigate their dimensions for $\delta$ in the range $1\le \delta \leq^{\lceil\frac{m}{2}\rceil+1}$. For non-narrow sense primitive BCH codes, weprovide two general formulas on their dimensions and give the dimensionsexplicitly in some cases. Furthermore, we settle the minimum distances of someprimitive BCH codes. We also explore the dimensions of the BCH codes of lengths$n=(q^m-1)/(q-1)$ and $n=q^m+1$ over finite fields.

Dimensions of three types of BCH codes over GF(q)